VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.6% → 99.7%

Time: 16.1s

Alternatives: 22

Speedup: 1.6×

Specification

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

C

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
end function

Java

public static double code(double F, double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Python

def code(F, B, x):
	return -(x * (1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
end

Wolfram

code[F_, B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (tan B))))
  (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))

C

double code(double F, double B, double x) {
	return -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
end function

Java

public static double code(double F, double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
}

Python

def code(F, B, x):
	return -(x * (1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))

Julia

function code(F, B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
end

Wolfram

code[F_, B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, \frac{-1}{\sin B}\right)\\ \mathbf{elif}\;F \leq 125000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, -\frac{x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, \frac{1}{\sin B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (tan B))))
   (if (<= F -2e+21)
     (fma t_0 x (/ -1.0 (sin B)))
     (if (<= F 125000000.0)
       (fma
        (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B))
        F
        (- (/ x (tan B))))
       (fma t_0 x (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / tan(B);
	double tmp;
	if (F <= -2e+21) {
		tmp = fma(t_0, x, (-1.0 / sin(B)));
	} else if (F <= 125000000.0) {
		tmp = fma((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), F, -(x / tan(B)));
	} else {
		tmp = fma(t_0, x, (1.0 / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / tan(B))
	tmp = 0.0
	if (F <= -2e+21)
		tmp = fma(t_0, x, Float64(-1.0 / sin(B)));
	elseif (F <= 125000000.0)
		tmp = fma(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), F, Float64(-Float64(x / tan(B))));
	else
		tmp = fma(t_0, x, Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+21], N[(t$95$0 * x + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 125000000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F + (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(t$95$0 * x + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e21

   1. Initial program 45.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{\tan B} \cdot x}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\tan B}\right)\right) \cdot x} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{\tan B}\right), x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\tan B}}\right), x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\tan B}}, x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1}}{\tan B}, x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      10. lower-/.f64 45.6

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{\tan B}}, x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \]

   4. Applied rewrites 45.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\right)} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\frac{-1}{\sin B}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\frac{-1}{\sin B}}\right) \]

      2. lower-sin.f64 99.7

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{-1}{\color{blue}{\sin B}}\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\frac{-1}{\sin B}}\right) \]

   if -2e21 < F < 1.25e8

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{F}{\sin B}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. clear-num N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\sin B}\right) \cdot F} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      9. div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \cdot F + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}, F, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, \frac{-x}{\tan B}\right)} \]

   if 1.25e8 < F

   1. Initial program 70.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{\tan B} \cdot x}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\tan B}\right)\right) \cdot x} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{\tan B}\right), x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\tan B}}\right), x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\tan B}}, x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1}}{\tan B}, x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right) \]

      10. lower-/.f64 70.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{\tan B}}, x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \]

   4. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\frac{1}{\sin B}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\frac{1}{\sin B}}\right) \]

      2. lower-sin.f64 99.7

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{1}{\color{blue}{\sin B}}\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\frac{1}{\sin B}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{-1}{\sin B}\right)\\ \mathbf{elif}\;F \leq 125000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, -\frac{x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{1}{\sin B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := t\_0 + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{\left(\frac{-1}{2}\right)}\\ t_2 := \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}\\ t_3 := \mathsf{fma}\left(\frac{t\_2}{\mathsf{fma}\left(B \cdot \left(B \cdot B\right), \mathsf{fma}\left(B \cdot B, 0.008333333333333333, -0.16666666666666666\right), B\right)}, F, -\frac{x}{\tan B}\right)\\ \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-26}:\\ \;\;\;\;t\_2 \cdot \left(F \cdot \frac{1}{\sin B}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-42}:\\ \;\;\;\;t\_0 + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{-1 + x}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1
         (+
          t_0
          (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) (/ -1.0 2.0)))))
        (t_2 (sqrt (/ 1.0 (fma F F 2.0))))
        (t_3
         (fma
          (/
           t_2
           (fma
            (* B (* B B))
            (fma (* B B) 0.008333333333333333 -0.16666666666666666)
            B))
          F
          (- (/ x (tan B))))))
   (if (<= t_1 -1000.0)
     t_3
     (if (<= t_1 -2e-26)
       (* t_2 (* F (/ 1.0 (sin B))))
       (if (<= t_1 5e-42)
         (+ t_0 (* (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (/ F B)))
         (if (<= t_1 20.0)
           (/ F (* (sin B) (sqrt (fma F F 2.0))))
           (if (<= t_1 INFINITY)
             t_3
             (/ (/ (- 1.0 (* x x)) (+ -1.0 x)) B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = t_0 + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), (-1.0 / 2.0)));
	double t_2 = sqrt((1.0 / fma(F, F, 2.0)));
	double t_3 = fma((t_2 / fma((B * (B * B)), fma((B * B), 0.008333333333333333, -0.16666666666666666), B)), F, -(x / tan(B)));
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = t_3;
	} else if (t_1 <= -2e-26) {
		tmp = t_2 * (F * (1.0 / sin(B)));
	} else if (t_1 <= 5e-42) {
		tmp = t_0 + (sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * (F / B));
	} else if (t_1 <= 20.0) {
		tmp = F / (sin(B) * sqrt(fma(F, F, 2.0)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = ((1.0 - (x * x)) / (-1.0 + x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ Float64(-1.0 / 2.0))))
	t_2 = sqrt(Float64(1.0 / fma(F, F, 2.0)))
	t_3 = fma(Float64(t_2 / fma(Float64(B * Float64(B * B)), fma(Float64(B * B), 0.008333333333333333, -0.16666666666666666), B)), F, Float64(-Float64(x / tan(B))))
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = t_3;
	elseif (t_1 <= -2e-26)
		tmp = Float64(t_2 * Float64(F * Float64(1.0 / sin(B))));
	elseif (t_1 <= 5e-42)
		tmp = Float64(t_0 + Float64(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * Float64(F / B)));
	elseif (t_1 <= 20.0)
		tmp = Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0))));
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(-1.0 + x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 / N[(N[(B * N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + B), $MachinePrecision]), $MachinePrecision] * F + (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[t$95$1, -1000.0], t$95$3, If[LessEqual[t$95$1, -2e-26], N[(t$95$2 * N[(F * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-42], N[(t$95$0 + N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 20.0], N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$3, N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -1e3 or 20 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < +inf.0

   1. Initial program 95.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{F}{\sin B}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. clear-num N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\sin B}\right) \cdot F} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      9. div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \cdot F + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}, F, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, \frac{-x}{\tan B}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}}}{\sin B}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      5. lower-fma.f64 98.6

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}, F, \frac{-x}{\tan B}\right) \]

   7. Applied rewrites 98.6%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}, F, \frac{-x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\color{blue}{B \cdot \left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)}}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{B \cdot \color{blue}{\left({B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) + 1\right)}}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\color{blue}{B \cdot \left({B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) + B \cdot 1}}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\color{blue}{\left(B \cdot {B}^{2}\right) \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)} + B \cdot 1}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\left(B \cdot {B}^{2}\right) \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) + \color{blue}{B}}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\color{blue}{\mathsf{fma}\left(B \cdot {B}^{2}, \frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, B\right)}}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\mathsf{fma}\left(\color{blue}{B \cdot {B}^{2}}, \frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, B\right)}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\mathsf{fma}\left(B \cdot \color{blue}{\left(B \cdot B\right)}, \frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, B\right)}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\mathsf{fma}\left(B \cdot \color{blue}{\left(B \cdot B\right)}, \frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, B\right)}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\mathsf{fma}\left(B \cdot \left(B \cdot B\right), \color{blue}{\frac{1}{120} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, B\right)}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\mathsf{fma}\left(B \cdot \left(B \cdot B\right), \color{blue}{{B}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), B\right)}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\mathsf{fma}\left(B \cdot \left(B \cdot B\right), {B}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}, B\right)}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\mathsf{fma}\left(B \cdot \left(B \cdot B\right), \color{blue}{\mathsf{fma}\left({B}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, B\right)}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\mathsf{fma}\left(B \cdot \left(B \cdot B\right), \mathsf{fma}\left(\color{blue}{B \cdot B}, \frac{1}{120}, \frac{-1}{6}\right), B\right)}, F, \frac{\mathsf{neg}\left(x\right)}{\tan B}\right) \]

      14. lower-*.f64 98.3

          \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\mathsf{fma}\left(B \cdot \left(B \cdot B\right), \mathsf{fma}\left(\color{blue}{B \cdot B}, 0.008333333333333333, -0.16666666666666666\right), B\right)}, F, \frac{-x}{\tan B}\right) \]

   10. Applied rewrites 98.3%

       \[\leadsto \mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\color{blue}{\mathsf{fma}\left(B \cdot \left(B \cdot B\right), \mathsf{fma}\left(B \cdot B, 0.008333333333333333, -0.16666666666666666\right), B\right)}}, F, \frac{-x}{\tan B}\right) \]

   if -1e3 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -2.0000000000000001e-26

   1. Initial program 94.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{F}{\sin B}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. clear-num N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\sin B}\right) \cdot F} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      9. div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \cdot F + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}, F, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, \frac{-x}{\tan B}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot \sqrt{\frac{1}{2 + {F}^{2}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{F}{\color{blue}{\sin B}} \cdot \sqrt{\frac{1}{2 + {F}^{2}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \]

      8. lower-fma.f64 75.4

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \]

   7. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 75.4%

      \[\leadsto \left(\frac{1}{\sin B} \cdot F\right) \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}} \]

   if -2.0000000000000001e-26 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 5.00000000000000003e-42

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}} \cdot \frac{F}{B} \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}} \cdot \frac{F}{B} \]

      8. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}} \cdot \frac{F}{B} \]

      11. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}} \cdot \frac{F}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \cdot \frac{F}{B} \]

      13. lower-/.f64 39.9

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 39.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}} \]

   if 5.00000000000000003e-42 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 20

   1. Initial program 90.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{F}{\sin B}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{F}{\sin B}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      6. clear-num N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      7. associate-/r/ N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \frac{1}{\sin B}\right) \cdot F} + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      9. div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \cdot F + \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}, F, \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

   4. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, \frac{-x}{\tan B}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot \sqrt{\frac{1}{2 + {F}^{2}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{F}{\color{blue}{\sin B}} \cdot \sqrt{\frac{1}{2 + {F}^{2}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \]

      7. unpow2 N/A

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \]

      8. lower-fma.f64 69.2

         \[\leadsto \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \]

   7. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 69.3%

      \[\leadsto \frac{F \cdot 1}{\color{blue}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}} \]

   if +inf.0 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 0.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 50.3

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.7%

      \[\leadsto \frac{-1 - x}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.9%

       \[\leadsto \frac{\frac{1 - x \cdot x}{x + -1}}{B} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{\left(\frac{-1}{2}\right)} \leq -1000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\mathsf{fma}\left(B \cdot \left(B \cdot B\right), \mathsf{fma}\left(B \cdot B, 0.008333333333333333, -0.16666666666666666\right), B\right)}, F, -\frac{x}{\tan B}\right)\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{\left(\frac{-1}{2}\right)} \leq -2 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \left(F \cdot \frac{1}{\sin B}\right)\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{\left(\frac{-1}{2}\right)} \leq 5 \cdot 10^{-42}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{\left(\frac{-1}{2}\right)} \leq 20:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{\left(\frac{-1}{2}\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\mathsf{fma}\left(B \cdot \left(B \cdot B\right), \mathsf{fma}\left(B \cdot B, 0.008333333333333333, -0.16666666666666666\right), B\right)}, F, -\frac{x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{-1 + x}}{B}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))